How do I move game pieces around a three dimensional board in a turn based game?
Represent the board using a more abstract notation. Essentially, you need a way to store co-ordinates in 3D space.

One way would be to place empties at all possible piece locations. Then you can snap to them. This is quite a limited approach. A better solution is to use a mathematical representation. For example, a sphere is very easily notated in ... spherical co-ordinates (two angles and a radius). Thus to snap a piece to a location is just a coordinate conversion issue. An easy way to think about this if you don't have experience with spherical coordinate systems is if you have a single piece at the top of the board, parent it to an empty in the middle of the board, and then rotate the empty. The piece will follow the surface of the sphere.

You will need python for this for the reasons pointed out by Quacksilber. I can think of ways to do it using animations and logic bricks but it will be a huge huge huge effort that will be very tedious.

I am quite interested in your game. Looks a bit like chess? Programming chess is never easy at the best of times (aka a flat board)